Although the maximum likelihood estimator enjoys asymptotic optimality prop erties, its finite-sample performance for a small or moderate sample size can be much improved when the Log-Likelihood is replaced by a Lq-Likelihood, which is motivated from a non-extensive measure of information (in contrast to the additive KullbackLcibler information).The properties of the new estimator, MLqE, are studied via asymptotic analysis and computer simulations.The behavior of the MLqE is char acterized by the degree of distortion q applied to the assumed model to amplify or diminish the low density values.When q is properly chosen for small and moderate sample sizes, the MLqE successfully trades bias for precision, resulting in a substan tial reduction of the mean squared error.When the sample size is large and q tends to 1, a necessary and sufficient condition to ensure a proper asymptotic normaiity and efficiency of MLqE is established.The advantage of the new estimation method is more clearly seen for higher dimensional estimations.(The talk is based on joint work with Davide Fcrrari.)